Dynamical Mordell-Lang and automorphisms of blow-ups

We show that if ϕ: X → X is an automorphism of a smooth projective variety and D ⊂ X is an irreducible divisor for which the set [d ∈ D: ϕⁿ(d) ∈2 D for some nonzero n] is not Zariski dense in D, then (X; ϕ) admits an equivariant rational fibration to a curve. As a consequence, we show that certain blow-ups (for example, blow-ups in high codimension) do not alter the finiteness of Aut(X), extending results of Bayraktar-Cantat. We also generalize results of Arnol'd on the growth of multiplicities of the intersection of a variety with the iterates of some other variety under an automorphism. These results follow from a nonreduced analog of the dynamical Mordell-Lang conjecture. Namely, let ϕ: X → X be an étale endomorphism of a smooth projective variety X over a field k of characteristic zero. We show that if Y and Z are two closed subschemes of X, then the set A_ϕ(Y;Z) = [n: ϕⁿ(Y) ⊆ Z] is the union of a finite set and finitely many residue classes; its modulus is bounded in terms of the geometry of Y .